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Chapter 4: Problem 43

In the following exercises, find three solutions to each linear equation. \(4 x-y=8\)

Short Answer

Expert verified
Three solutions are: (0, -8), (2, 0), and (3, 4).

Step by step solution

01

- Understand the Linear Equation

The given linear equation is in the form of Ax + By = C. Here, it is written as: 4x - y = 8
02

- Rewrite the Equation in Terms of y

To find solutions, rewrite the equation in terms of y: 4x - y = 8 Add y to both sides: 4x = y + 8 Subtract 8 from both sides: y = 4x - 8
03

- Choose Values for x

To find coordinates (x, y), choose different values for x and solve for y.Example 1: Choose x = 0y = 4(0) - 8 = -8 So, one solution is (0, -8).Example 2: Choose x = 2y = 4(2) - 8 = 0 So, another solution is (2, 0).Example 3: Choose x = 3y = 4(3) - 8 = 4 The third solution is (3, 4).
04

- Verify the Solutions

Substitute each pair back into the original equation to ensure they satisfy 4x - y = 8. For (0, -8): 4(0) - (-8) = 8 For (2, 0): 4(2) - 0 = 8 For (3, 4): 4(3) - 4 = 8 Each pair satisfies the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Equations
Linear equations are a fundamental concept in algebra. They represent a straight line when graphed on a coordinate plane. A basic form of a linear equation is Ax + By = C, where A, B, and C are constants and x and y are variables. From the exercise, we see that the linear equation is given as: 4x - y = 8. This specific equation can be manipulated by algebraic techniques so that we can find pairs of (x, y) that satisfy the equation.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for a specific variable. For the problem 4x - y = 8, we need to express y in terms of x. Here are the steps:
  • Add y to both sides: 4x = y + 8
  • Subtract 8 from both sides: y = 4x - 8
Now, we have y = 4x - 8. This rearrangement makes it easier to pick values for x and then solve for y. Let's say we choose x = 0:
y = 4(0) - 8 = -8. Thus, one solution is (0, -8). By repeating this process with different values of x, such as x = 2 and x = 3, we obtain (2, 0) and (3, 4), respectively.
Solution Verification
After finding solutions, it's always a good practice to verify them. Verification ensures the correctness of our solutions. To verify, substitute each pair back into the original equation 4x - y = 8 and check if it holds true:
  • For (0, -8): 4(0) - (-8) = 8
  • For (2, 0): 4(2) - 0 = 8
  • For (3, 4): 4(3) - 4 = 8
As you can see, each of the solutions satisfies the original equation. This confirms that (0, -8), (2, 0), and (3, 4) are all correct solution pairs. Verifying each step helps avoid errors and ensures a deeper understanding of linear equations.

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